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Beirut Arab University
Computer Engineering Program
Spring 2015
Digital Signal Processing (COME 384)
Instructor: Prof. Dr. Hamed Nassar
Fast Fourier Transforms FFT
 Textbook:
◦ V. Ingle and J. Proakis, Digital Signal Processing
Using MATLAB 3rd Ed, Cengage Learning, 2012
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Dr. Hamed Nassar, Beirut Arab Univ

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DSP Windows
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Prof. H. Nassar, BAU

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DSP Windows
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Prof. H. Nassar, BAU

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DSP Windows MATLAB code
 n=0:9 %stem ten samples
 RectWin=ones(1,10)
 sub1=subplot(4,1,1); stem(n, RectWin);grid
 title('Rectangular Window');
 TriWin=1-abs(length(n)-1-2*n)/(length(n)-1)
 sub2=subplot(4,1,2); stem(n, TriWin);grid
 title('Triangular Window');
 HannWin=0.5-0.5*cos(2*pi*n/(length(n)-1)))
 sub3=subplot(4,1,3); stem(n, HannWin);grid
 title('Hanning Window');
 HammWin=0.54-0.46*cos(2*pi*n/(length(n)-1))
 sub4=subplot(4,1,4); stem(n, HammWin);grid
 title('Hamming Window');
 linkaxes([sub1,sub2,sub3,sub4], 'y')%Use same y scale for
all subs Prof. H. Nassar, BAU

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Constructing an infinite, periodic sequence from a finite
one, and vv
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Prof. H. Nassar, BAU

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Circular Shift Example: 4 left
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Prof. H. Nassar, BAU

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Circular convolution
 A linear convolution between two N-point sequences
results in a longer sequence (length=N+N-1, with indices:
0, 1, …, 2N-2).
 But we have to restrict our interval to 0 ≤ n ≤ N − 1,
therefore instead of linear shift, we should consider the
circular shift.
 A convolution operation that contains a circular shift, as
shown below, is called the circular convolution and is
given by
Prof. H. Nassar, BAU

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Circular Convolution Example: Time domain approach (watch for circular
folding)
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Prof. H. Nassar, BAU

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Circular Convolution Example: Frequency domain approach
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Prof. H. Nassar, BAU

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FFT
 The DFT introduced earlier is the only transform that is
discrete in both the time and the frequency domains, and
is defined for finite-duration sequences.
 Although it is a computable transform, its straightforward
computation is very inefficient, especially when the
sequence length N is large.
 In 1965 Cooley and Tukey showed a procedure to
substantially reduce the amount of computations involved
in the DFT.
 This led to the explosion of other efficient algorithms
collectively known as fast Fourier transform (FFT)
algorithms.
 We will study one such algorithms in detail.
 One concept used that the book does not speak about
explicitly, but it is there (seen the previous example on
circular convolution) is circular folding.
Prof. H. Nassar, BAU

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FFT
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Prof. H. Nassar, BAU

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Example
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Prof. H. Nassar, BAU

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Divide and Conquer Algorithm
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Prof. H. Nassar, BAU

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Implementation
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Prof. H. Nassar, BAU

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Implementation
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Prof. H. Nassar, BAU

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EXAMPLE 5.21 N = 15, L = 3, M =5
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Prof. H. Nassar, BAU

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EXAMPLE 5.21 N = 15, L = 3, M =5
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Prof. H. Nassar, BAU

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EXAMPLE 5.21 N = 15, L = 3, M =5
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Prof. H. Nassar, BAU

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Reminder: Classical DFT MATLAB
script
 % This is a DFT program without loops
 clear all; %Write input sequence below
 nx=0:5; x=[0 1 7 3 1 2] %index vector nx, then input signal
x
 N=length(x);
 wN=exp(-2j*pi/N) %Base
 E=[0:N-1]' *[0:N-1] %Prepare structure of matrix W
 W=wN.^E %Matrix W=Base^Matrix (note the dot .)
 Y=x*W %DFT = input sequence * Matrix
 %To test, we'll obtain x(n=2)=1/N sum_k=0^N-1 Y
e^(j*2*pi*n*k)
 n=2
 k=nx;
 g=exp(2j*pi*n*k/N)
 G=1/N*(Y*g.') Prof. H. Nassar, BAU

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FFT Example: N = 6, M=4, L=2
 x = {3, 1, 0, 2, 4, 1,3, 2} => x = {3, 1, 0, 2, 4, 1, 3, 2}
 This has DFT: 10.0, 0.5-2.6i, 2.5+2.6i, 2.0, 2.5-2.6i,
0.5+2.6i
 Always remember that:
◦ the 1D input vector x is first converted (reshaped) into a
row-major matrix xm of size LxM
◦ the corresponding output DFT vector X is obtained as a
column-major matrix X of size LxM, and thus has to be
converted (reshaped) back into a 1D vector.
◦ The best summary of the Divide and Conquer FFT algo:
Prof. H. Nassar, BAU

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 clear all; % FFT using Divide & Conquer without loops
 x =[3 1 0 2 4 1 3 2];L=2;M=4; %%Write input sequence and your choice for L and M
 N=length(x)%N can be obtained auto, but M,L u should spec
 x=reshape(x, [L,M]) %Create an LxM matrix (col maj) from x
 wM=exp(-2j*pi/M) %Base M
 EM=[0:M-1]' *[0:M-1]%Prepare structure of matrix W: MxM
 WM=wM.^EM %Raise elements (note .) of Matrix to Base
 WLMF=x*WM %Obtain F matrix
 wN=exp(-2j*pi/N) %Base N
 ELM=[0:L-1]'*[0:M-1] %Prepare structure of twiddle matrix:LxM
 WLMT=wN.^ELM %Twiddle: Raise elements (.) of Matrix to Base
 WLMG=WLMF.*WLMT %Modify elements (note .) of F by Twiddle
 wL=exp(-2j*pi/L) %Base L
 EL=[0:L-1]'*[0:L-1] %Prepare structure of matrix W:LxL
 WL=wL.^EL %Raise elements (note .) of Matrix to Base
 X=WL*WLMG %Final DFT
 %To test, obtain x(n)=1/N sum_k=0^N-1 Y e^(j*2*pi*n*k) for some n
 n=6; k=0:N-1; %MATLAB: u involve vec in an op, u get vec
 XX=[X(1,:) X(2,:)] %Unfold matrix into a vector
 g=exp(2j*pi*n*k/N) %Note in IDFT sign of e is +ve
 G=1/N*(XX*g.') %IDFT for the given n. We could obtain whole x
Prof. H. Nassar, BAU

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5
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Prof. H. Nassar, BAU